Muga G. Time in Quantum Mechanics

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2 Time Dependent Schr

odinger Equation 21

which is the non-relativistic wave equation, again obtained without introducing

operator-valued time or momentum.

2.4 The Single Photon and Two Photon Wave Functions

The photon wave function concept really comes into its own when solving problems

involving photon–photon correlations. Then, as is explained in [8], the two photon

wave function

(2)

, t

; r

, t

) ≡0|

E(r

, t

)

E(r

, t

)|Ψ  (2.27)

is the subject of interest. Under some conditions this may be written in terms of

single photon wave functions, as in the case of two photon cascade discussed below.

Some of the calculational details will be given since the physics (and the devil) is in

the details.

Consider ﬁrst the single photon wave function. From Eqs. (2.3) and (2.4) and

ignoring polarization, we ﬁnd

0|

E(r, t)|ψ

=





2ε



(ν

)

1/2

−iν

ik·(r−r

)

(ν

−ω) +iΓ/2

. (2.28)

We now evaluate this function by converting the sum into an integral. The φ- and θ-

integrations can be carried out by choosing a coordinate system in which the vector

r−r

points along the z-axis. We then carry out the integration over |k|by evaluating

the density of states and matrix elements at resonance. We are left with the integral



∞

−∞

dν

−iν

t+iν

Δr/c

(ν

−ω) +iΓ/2

which is evaluated via contour methods and where Δr =|r − r

| is the distance

from the atom located at position r

to the detector. For t <Δr/c, the contour lies

in the upper half-plane and if t >Δr/c, in the lower half-plane. On performing the

integration, we ﬁnd

0|

E(r, t)|ψ

=

Δr



t −

Δr



−i(t−

Δr

)(ω−iΓ/2)

, (2.29)

where Θ is a unit step function and E is an overall constant with the units of electric

ﬁeld.

Next we consider the problem of “interrupted” emission, see Fig. 2.3. The ﬁrst

photon, associated with the a ↔ b transition, is described in the long time limit by

our “old friend”

22 M.O. Scully

Fig. 2.3 Figure illustrating

decay of atom excited to state

a at a rate γ

to non-decaying

level b. Upon detection of

a → b photon, population in

level b is transferred to b



means of an external ﬁeld

indicated by wavy line.Level



decays to c at rate γ

|γ =



a,k

−ik·r

(ω

−c|k|) −iγ

 . (2.30)

Likewise the second photon, associated with the b



→ c transition, is given in

the long time limit by

|φ=



b,q

−i(q·r−cqt

)

(ω

−c|q|) −iγ

 , (2.31)

where t

is the time of detection of γ photon and the transfer from b → b



Using (2.30) and (2.31), it is easy to calculate the two photon wave function

(2)

, t

; r

, t

) as deﬁned by (2.27). We ﬁnd

(2)

, t

; r

, t

) = ψ

, t

)ψ

, t

) + ψ

, t

)ψ

, t

) , (2.32)

where

, t

) =

Δr



−

Δr



−γ

−

Δr

)

−iω

−

Δr

)

, (2.33)

and

, t

) =

Δr



−t

−

Δr



−γ

−t

−

Δr

)

−iω

−t

−

Δr

)

, (2.34)

where i = 1, 2 designates the detector positions.

2.5 Conclusions

One motive for this chapter is to show that the time appearing in the classical

Maxwell equations is the same as the time parameter which appears in the TDSE.

Thus, the times appearing in classical mechanics and electrodynamics and quantum

mechanics are all the same.

Another motivation involves the deﬁnition of the photon wave function in terms

of the electric and magnetic operators as

2 Time Dependent Schr

odinger Equation 23

(r, t) =0|

E(r)|Ψ (t), (2.35)

and

(r, t) =0|

H(r)|Ψ (t). (2.36)

Equations (2.6) and (2.11) are the analog of the matter wave probability

amplitudes

Ψ (r, t) =0|

ψ(r)|Ψ (t) (2.37)

discussed at length in Sect. 2.1.

As explained in [3], the discussion of the proceeding paragraph serves to put the

nice question of Kramers [9] in perspective. Speciﬁcally, Kramers asks,

When in 1924 De Broglie suggested that material particles should show wave phenom-

ena ...such a comparison was of great heuristic importance. Now that wave mechanics has

become a consistent formalism one could ask whether it is possible to consider the Maxwell

equations to be a kind of Schr

odinger equation of light particles ...?

Kramers answers his question in the negative, he says,

Thus it is natural to ask what are the φ’s for photons? Strictly speaking there are no such

wave functions! One may not speak of particles in a radiation ﬁeld in the same sense as

in the elementary quantum mechanics of systems of particles as used in the last chapter.

The reason is that the wave equation ...solutions of Schr

odinger’s time dependent wave

function corresponding to an energy E

have a circular frequency ω

=+E

/, while the

monochromatic solutions of the wave equation have both ±ω

In other words, Kramers is saying that “the real electric wave has both exp(−iν

and exp(iν

t) parts while the matter wave has only exp(−iν

t) type terms.”

However, from the quantum optical perspective, we see that the photon wave

functions (2.35) and (2.36) and the matter wave function (2.37) are identical in spirit.

An earlier discussion of the importance of the analytical (positive frequency) signal

in this context was given by Sudarshan [10].

The present measurement theory, “a-photon-is-what-a-photodetector-detects”

point-of-view is discussed further in [3]. We have also included in Sect. 2.4 a

detailed photon–photon correlation analysis [8] for the convenience of the reader.

Acknowledgments I would like to thank R. Arnowitt, C. Summerﬁeld, and S. Weinberg for useful

and stimulating discussions. This work was supported by the Robert A. Welch Foundation grant

number A-126 and the ONR award number N00014-07-1-1084.

References

1. M. Scully, J. Phys. Conf. Ser. 99, 012019 (2008); Wilhelm and Else Heraeus-Seminar (no. 395)

in Honor of Prof. M. Kleber (Blaubeuren, Germany, Sept 2007) 15

24 M.O. Scully

2. K. Chapin, M. Scully, M.S. Zubairy, in Frontiers of Quantum and Mesoscopic Thermodynam-

ics Proceedings (28 July–2 August 2008), Physica E, to be published 15

3. M. Scully, M.S. Zubiary, Quantum Optics (Cambridge Press, Cambridge, 1997) 16, 17, 23

4. S. Weinberg, The Quantum Theory of Fields I (Cambridge Press, Cambridge, 2005) 19

5. S. Schweber, An Introduction to Relativistic Quantum Field Theory (Harper and Row, New

York, 1962) 19

6. R. Good, T. Nelson, Classical Theory of Electric and Magnetic Fields (Academic Press, New

York, 1971) 19

7. R. Oppenheimer, Phys. Rev. 38, 725 (1931) 19

8. M. Scully, in Advances in Quantum Phenonema, E. Beltrametti, J.-M. L

evy-Leblond (eds.)

(Plenum Press, New York, 1995) 21, 23

9. H. Kramers, Quantum Mechanics (North Holland, Amsterdam, 1958) 23

10. E.C.G. Sudarshan, Phys. Rev. Lett. 10, 277 (1963) 23

Chapter 3

Post-Pauli’s Theorem Emerging Perspective

on Time in Quantum Mechanics

Eric A. Galapon

3.1 Introduction

In a Hilbert space setting, Pauli’s well-known theorem asserts that no self-adjoint

operator exists that is conjugate to a semibounded or discrete Hamiltonian [55].

Pauli’s argument goes as follows. Assume that there exists a self-adjoint operator

T conjugate to a given Hamiltonian H, that is,

[

T, H

]

= iI; such an operator

conjugate to the Hamiltonian is known as a time operator. Since T is self-adjoint,

the operator U

= exp(−iεT) is unitary for all real number ε.Nowifϕ

is an

eigenvector of H with the eigenvalue E, then, according to Pauli, the conjugacy

relation

[

T, H

]

= iI implies that T is a generator of energy shifts so that HU

(E + ε)ϕ

E+ε

; this means that H has a continuous spectrum spanning the entire real

line because ε is an arbitrary real number. Hence, the ‘inevitable’ conclusion that if

the Hamiltonian is semibounded or discrete no self-adjoint time operator T will exist

to satisfy

[

T, H

]

= iI. A modern reading of Pauli’s theorem is that the conjugacy

relation [T, H] = iI implies that the pair T and H form a system of imprimitivities

over the entire real line, so that when H is semibounded or discrete T cannot be

self-adjoint [59].

It is Pauli’s theorem that has distilled the idea that self-adjointness and conjugacy

of a time operator for a semibounded or discrete Hamiltonian cannot be imposed

simultaneously [29, 43, 56, 59, 53, 60, 44]. Since quantum observables are pos-

tulated to be self-adjoint operators in the earlier days of quantum mechanics [63],

the non-existence of self-adjoint time operator has been interpreted to mean that

time is not a dynamical observable but a mere parameter marking the evolution of

a quantum system [58, 36, 54, 7, 2]. However, it is likewise widely recognized that

time acquires dynamical signiﬁcance in questions involving the occurrence of an

event [59, 8, 52, 12] – when a nucleon decays [15] or when a particle arrives at a

given spatial point [51, 39] or when a particle emerges from a potential barrier [48].

E.A. Galapon (B)

Theoretical Physics Group, National Institute of Physics, University of the Philippines, Diliman,

Quezon City, Philippines, eric.galapon@up.edu.ph

Galapon, E.A.: Post-Pauli’s Theorem Emerging Perspective on Time in Quantum Mechanics. Lect.

Notes Phys. 789, 25–63 (2009)

DOI 10.1007/978-3-642-03174-8

 Springer-Verlag Berlin Heidelberg 2009

26 E.A. Galapon

Moreover, there is the time–energy uncertainty principle, a reasonable interpretation

of which requires more than a parametric treatment of time [8, 1, 3, 9, 16, 17, 40–42].

This opposing view on time in quantum mechanics precipitated to what is now

known as the quantum time problem.

Pauli’s theorem has been so ingrained into the physicist’s psyche that it stiﬂed

serious sustained research on the quantum dynamical aspect of time until quite

recently. The realization that quantum observables are not necessarily self-adjoint

but may be non-self-adjoint as ﬁrst moments of positive operator-valued measures

(POVM) has brought a resurgence of interest on the quantum time problem. The

introduction of POVM observables has opened up the possibility of entertaining

non-self-adjoint, conjugate time operators as quantum observables, because such

operators may be ﬁrst moments of certain POVMs [59, 2, 10]; the quantized free

time of arrival operator is an example of such a non-self-adjoint operator conju-

gate to the free Hamiltonian [14]. Since Pauli’s theorem has been understood to

mean that T and H are each other’s generator of translations in their respective

spectral measures, it has been the belief that a time operator must inevitably be

non-self-adjoint for semibounded Hamiltonians and must cardinally be covariant

and a POVM observable [29, 59, 60, 2, 8, 9, 14, 4]. Now covariance requires that a

time operator must at least have a completely continuous spectrum. This altogether

denies the possibility of constructing self-adjoint time operators that are bounded

and compact for semibounded Hamiltonians.

However, while a sustained development in the dynamical aspect of time under

the motivation of POVM observables is in progress, an unexpected development

has emerged: a counter example to Pauli’s theorem in the Hilbert space formulation

of quantum mechanics is constructed, exposing the subtle assumptions that go into

Pauli’s arguments that cannot be sustained. In [19] we have shown the consistency

of a bounded and self-adjoint operator conjugate to a discrete and semibounded

Hamiltonian, contrary to Pauli’s claim. There we have explicitly shown that the

quantized classical free time of arrival for a spatially conﬁned particle is self-adjoint,

compact, and conjugate to the Hamiltonian in a non-dense subspace of the Hilbert

space. This in effect has demonstrated that the non-self-adjointness of the same

formal quantized operator for a particle in the real line has nothing to do with the

semiboundedness of the free Hamiltonian, again, contrary to expectations due to

Pauli’s theorem. The existence of such self-adjoint time operators has opened up

a new window through which the quantum time problem can be viewed from a

different perspective.

In this chapter, we synthesize the progress that we have made since the appear-

ance of [19], in particular, to our solution to the quantum time of arrival problem

in the interacting case [50, 4, 49, 21]. Our solution consists of generalizing the

time of arrival for a spatially conﬁned particle in [19] under more general boundary

conditions and in the presence of an interaction potential. This generalization led to

the introduction of the conﬁned quantum time of arrival (CTOA) operators, which

are both conjugate and self-adjoint [24, 25, 22]. The dynamical behaviors of the

eigenfunctions of the CTOA operators lead to a coherent theory of quantum arrival

in one dimension that can yield both time of arrival distributions and at the same

3 Post-Pauli’s Theorem 27

give a mechanism for the appearance of particle at the moment of its arrival [26, 23].

The resulting theory of quantum arrival invites us to reconsider our beliefs on time

operators and on the role of time in quantum measurement theory.

3.2 Quantum Canonical Pairs

3.2.1 Canonical Pairs in Hilbert Spaces

We cannot start to appreciate the signiﬁcance of the counter example to Pauli’s the-

orem without a clear understanding of the properties of a canonical pair in a Hilbert

space. To the physicist, a canonical pair is a pair of operators (Q, P) satisfying the

canonical commutation relation, [Q, P] = iI, (CCR), but a quantum canonical pair

is much more elaborate than that. Failure to recognize its ramiﬁcations can lead to

unwarranted claims and conclusions regarding the properties of such a pair [11].

Let H be the system Hilbert space, which we assume to be inﬁnite dimensional. If

we seek a pair of operators in H, Q and P, with respective domains D

and D

satisfying the CCR, then two facts must be recognized:

1. No pair (Q, P) exists to satisfy the CCR in the entire Hilbert space H.

That is, there are no Q and P such that [Q, P]ϕ = iϕ for all ϕ in H,or

[Q, P] = iI

, where I

is the identity in H. A pair (Q, P) can at most satisfy

the CCR in a proper subspace, D

,ofH; that is, the relation [Q, P]ϕ = iϕ

holds only for all those ϕ in D

, where D

is always smaller than H. Thus a

canonical pair in a Hilbert space is a triple C(Q, P; D

) – a pair of Hilbert space

operators, Q and P, together with a non-trivial, proper subspace D

of H, which

we refer to as the canonical domain. The canonical domain may or may not be

dense;

it may not even be invariant under either Q or P. These subtle properties

of the canonical domain generally forbid us from acting arbitrarily with Q and

P on D

. Failure to pay attention to these small details can lead to erroneous

generalizations; for example, the conclusion that the spectra of Q and P are the

entire real line because they satisfy the CCR (see, for example, [11]) requires, at

least, the canonical domain be dense and invariant under Q and P. When even just

one of these conditions is not satisﬁed, the conclusion no longer holds. In general

the commutator domain, D

com

= D

∩D

, the domain in which (QP−PQ)is

deﬁned in the Hilbert space, does not coincide with the canonical domain. That

is, the canonical commutation relation [Q, P]ϕ = iϕ does not hold in general

for arbitrary elements of D

com

but only for certain elements of a subset D

com

2. There are canonical pairs in the same Hilbert space that do not share the same

properties.

A subspace D of H is dense if the only vector orthogonal to all elements of D is the zero vector.

28 E.A. Galapon

This means that, for a given Hilbert space H, we can ﬁnd different pairs of oper-

ators (Q

, P

) acting in H , together with corresponding subspaces D

, such that we

have the canonical pairs C(Q

, P

; D

). The pairs (Q

, P

) and (Q



, P



) may be

different in the sense that there is no unitary operator U such that Q



= UQ

−1

and P



= UP

−1

. For such cases, the pairs C(Q

, P

; D

) will have different

spectral properties, e.g., one pair may be self-adjoint, another non-self-adjoint. Also

it is possible that for a given operator Q there may be several distinct P

’s with

corresponding subspaces D

– that is, P

= P



and D

= D



for k = k



– such

that for every k we have the canonical pair D(Q, P

; D

). Of course for a given P

and D

, such that we have the canonical pair C(Q, P; D

), we can also ﬁnd another

operator P



= P + F with [Q, F]ϕ = 0 for all ϕ in D

such that we have another

canonical pair C



(Q, P



; D

). But we mean more than that: For P

= P



there may

not be an F such that P



= P

+F. We will illustrate later how these different cases

may arise in certain physical systems.

3.2.2 Classiﬁcation of Hilbert Space Solutions to the CCR

For a given Hilbert space H, we refer to a canonical pair C(Q, P; D

), with Q and

P both operators in H, as a solution to the CCR.

Solutions split into two major

categories, according to whether the canonical domain D

is dense or not. We shall

say that a canonical pair is of dense-category if the corresponding canonical domain

is dense; otherwise, it is of closed category. Solutions under these categories further

split into distinct classes of unitary equivalent pairs, and each class will have its

own set of properties. Under such categorization of solutions, the CCR in a given

Hilbert space H assumes the form [Q, P] ⊂ iP

, where P

is the projection

operator onto the closure

of the canonical domain D

. If the pair C(Q, P; D

)

is of dense category, then the closure of D

is just the entire H, so that P

is the

identity I

of H. In fact, we are considering a more general solution set to the CCR

than has been considered so far. The traditional reading of the CCR in H is the form

[Q, P] ⊂ iI

, which is just the dense category.

It can be shown that the canonical and commutator domains coincide for dense

category canonical pairs, that is, D

= D

com

; on the other hand, the canonical

domain is smaller than and contained in the commutator domain for closed category

canonical pairs, that is, D

⊂ D

com

[27]. Since only the dense category solutions

have been the subject of investigations so far, we have gotten used to dealing with

canonical pairs in the entire commutator domain and may feel suspicious with the

We avoided to use the more mathematically accurate term representation infavoroftheterm

solution. The reason is that representation carries an extra connotation in physics in which it usu-

ally implies equivalence. For example, we have position and momentum representations, and we

know that these two representations are equivalent so that it does not matter which one we use in

describing our system. In fact, the use of phrases such as representations of the Heisenberg pair

in physics literature has added to the confusion on the exact nature of quantum canonical pairs, in

particular, giving the impression that different canonical pairs have similar properties.

3 Post-Pauli’s Theorem 29

closed category solutions. However, the conﬁned time of arrival operators, together

with their Hamiltonians, form such a class of canonical pairs, and they, as we will

see later, have an unambiguous physical origin.

3.2.3 Is There a Preferred Solution to the CCR?

We discussed above that for a given Hilbert space H there are numerous solutions

to the canonical commutation relation that do not necessarily share the same prop-

erties. So is there a preferred solution to the CCR? Should we accept only solutions

of dense or closed category of a speciﬁc class? Let us see how different solutions

may arise in a given Hilbert space and see how each solution may represent different

systems.

Let us consider the well-known position and momentum operators in three differ-

ent conﬁguration spaces: The entire real line, Ω

= (−∞,∞); the bounded segment

of the real line, Ω

= (0, 1); and the half line Ω

= (0, ∞). Quantum mechanics

in each of these happens in the Hilbert spaces H

= L

(Ω

), H

= L

(Ω

), and

= L

(Ω

), respectively. The position operators, Q

,inH

, for all j = 1, 2, 3,

arise from the fundamental axiom of quantum mechanics that the propositions for

the location of an elementary particle in different volume elements of Ω

are com-

patible (see Jauch [45] for a detailed discussion for Ω

, which can be extended to

and Ω

). They are self-adjoint and are given by the operators (Q

ϕ)(q) = qϕ(q)

for all ϕ in the domain D



ϕ ∈ H

: Q

ϕ ∈ H



. Note that Q

and Q

are

both unbounded, while Q

is bounded.

Now each of the conﬁguration spaces, Ω

, Ω

, and Ω

, has an identifying prop-

erty. Ω

is fundamentally homogeneous – points there are physically indistinguish-

able. On the other hand, Ω

and Ω

are not homogeneous, the boundaries being the

distinguishing factor. However, their inhomogeneities are not the same, their number

of boundaries being different. These properties can be expressed mathematically in

terms of the respective representation of translation in each of these conﬁguration

spaces. Translation in Ω

is isomorphic to the additive group of real numbers; in

, to the group of rotations of the circle; in Ω

, to the semigroup of additive pos-

itive numbers. Thus in H

and H

there are one-parameter unitary operators U

(s),

(s) representing translations in H

and H

, respectively. And in H

there is a

completely one-parameter semigroup U

(s) representing translations. If we deﬁne

the momentum operator as the generator of translation in the conﬁguration space,

then the momentum operator in H

is the operator P

deﬁned on all vectors ϕ for

which the limit lim

s→0

(is)

−1

(s) −I

)ϕ = P

ϕ exists. Explicitly, it is given by

ϕ)(q) =−iϕ



(q).

In each H

, there exists a dense common subspace D

of Q

and P

, which is

invariant under Q

and P

, for which we have the canonical pair C

, P

; D

The C

’s are of the same dense category, but they belong to different classes: Q

and

are both self-adjoint, having absolutely continuous spectra spanning the entire

real line Re and forming a system of imprimitivities in Re, and their restrictions

30 E.A. Galapon

in D

are essentially self-adjoint. Q

is self-adjoint with an absolutely continuous

spectra in (a, b), and its restriction in D

is essentially self-adjoint; P

is self-adjoint

with a pure point spectrum, but its restriction in D

is not essentially self-adjoint.

is self-adjoint with an absolutely continuous spectra in (0, ∞), and its restriction

in D

is essentially self-adjoint; P

is maximally symmetric and non-self-adjoint,

thus without any self-adjoint extension. These varied properties of the position and

momentum canonical pairs are obviously the consequences of the underlying prop-

erties of their respective conﬁguration spaces.

So is there a preferred solution to the CCR? Recall that there is only one separable

Hilbert space; that is, all separable Hilbert spaces are isomorphically equivalent to

each other, so that there are unitary operations transforming one Hilbert space to

another. The three Hilbert spaces, H

, H

, and H

, are separable, and hence can be

transformed to a common Hilbert space H

, together with all the operators in them,

including their respective position and momentum operators. The canonical pairs,

, C

}, are then solutions of the CCR in the same Hilbert space H

.Andwe

have seen that they are of dense category solutions, but of different classes – and,

most important, they represent different physical systems. If we look at the diverse

properties of the above C

’s, we can see that these properties are reﬂections of the

fundamental properties of the underlying conﬁguration spaces of their respective

physical systems.

It is then misguided to prefer one solution of the CCR over the rest or to require

a priori a particular category of a speciﬁc class of a solution without a proper con-

sideration of the physical context against which the solution is sought. For example,

if we insist that only canonical pairs forming a system of imprimitivities over the

real line are acceptable, then, within the context of position–momentum pairs, we

are imposing homogeneity in all conﬁguration spaces. But why impose the homo-

geneity of, say, Ω

in intrinsically inhomogeneous conﬁguration spaces like Ω

and

From the position–momentum example, it can be concluded that the set of prop-

erties of a speciﬁc solution to the CCR is consequent to a set of underlying fun-

damental properties of the system under consideration, or to the basic deﬁnitions

of the operators involved, or to some fundamental axioms of the theory, or to some

postulated properties of the physical universe. That is, a speciﬁc solution to the CCR

is canonical in some sense, i.e., of a particular category and of a particular class. It is

conceivable to impose that a given pair be canonical as a priori requirement based,

say, from its classical counterpart, but not without a deeper insight into the underly-

ing properties of the system. In other words, we do not impose in what sense a pair is

canonical if we do not know much, we derive in what sense instead. Furthermore, if

a given pair is known to be canonical in some sense, then we can learn more about

the system or the pair by studying the structure of the sense the pair is canonical

[19].

We can appreciate this statement further by noting that ﬁnding solutions to the

CCR in a Hilbert space is akin to solving a differential equation in which there is

no preferred solution until appropriate boundary or initial condition is imposed.

Also as in differential equations where the imposed conditions may not admit

Muga G. Time in Quantum Mechanics - Vol. 2

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